Weakstrong Uniqueness Property for the Compressible Flow

J. Differential Equations 255 (2013) 12331253

Contents lists available at SciVerse ScienceDirect

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eakstrong uniqueness property for the compressible owf liquid crystals

ong-Fu Yang a,c,, Changsheng Dou b,c, Qiangchang Ju c

epartment of Mathematics, College of Sciences, Hohai University, Nanjing 210098, Jiangsu Province, PR Chinachool of Statistics, Capital University of Economics and Business, Beijing 100070, PR Chinanstitute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, PR China

r t i c l e i n f o a b s t r a c t

ticle history:ceived 30 March 2012vised 12 April 2013ailable online 28 May 2013

ywords:uid crystalmpressible hydrodynamic owlative entropyeakstrong uniqueness

Weakstrong uniqueness property in the class of nite energyweak solutions is established for two different compressible liquidcrystal systems by the method of relative entropy. To overcome thediculties caused by the molecular direction with inhomogeneousDirichlet boundary condition, new techniques are introduced tobuild up the relative entropy inequalities.

2013 Elsevier Inc. All rights reserved.

Introduction

In physics, liquid crystals are states of matter which are capable of ow and in which the molecularrangements give rise to a preferred direction. Nematic liquid crystals are aggregates of moleculeshich posses same orientational order and are made of elongated, rod-like molecules. The continuumeory of compressible (or incompressible) liquid crystals was rst developed by Ericksen [5] andslie [15] during the period of 1958 through 1968. Since then there have been remarkable researchvelopments in liquid crystals from both theoretical and applied aspects.When the uid is an incompressible, viscous uid, Lin [17] rst derived simplied EricksenLeslieuations modeling liquid crystal ows in 1989. Subsequently, the global existence of weak solutionsith large initial data was proved under the condition that the orientational conguration belongsH2, and the global existence of classical solutions was also obtained if the viscosity coecient is

rge enough in the three-dimensional spaces by Lin and Liu [1820]. While the uid is allowed to

Corresponding author at: Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, Jiangsuovince, PR China.E-mail addresses: [email protected] (Y.-F. Yang), [email protected] (C. Dou), [email protected] (Q. Ju).

22-0396/$ see front matter 2013 Elsevier Inc. All rights reserved.

tp://dx.doi.org/10.1016/j.jde.2013.05.011

1234 Y.-F. Yang et al. / J. Differential Equations 255 (2013) 12331253

beto

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wde

wF

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an

w

didecompressible, the corresponding EricksenLeslie system becomes more complicated and dicultstudy mathematically due to the compressibility.Our goal of this paper is to establish weakstrong uniqueness property for the compressible ow ofuid crystals. To begin with, we consider the following simplied version of EricksenLeslie system:

t + div(u) = 0,(u)t + div(u u) + P = u div

(d d

(1

2|d|2 + F (d)I3

)),

dt + u d = (d f(d)),

(1.1)

here 0 denotes the density, u R3 the velocity, d R3 the direction eld for the averagedacroscopic molecular orientations, and P () = a the pressure with constant a > 0 and the adia-tic exponent 1. The positive constants , , denote the viscosity, the competition betweennetic energy and potential energy, and the microscopic elastic relation time for the molecular orien-tion eld, respectively. The symbol denotes the Kronecker tensor product, I3 is the 3 3 identityatrix, and d d denotes the 3 3 matrix whose i j-th entry is xid, x jd. Indeed,

d d = (d)Td,

here (d)T stands for the transpose of the 33 matrix d. The vector-valued smooth function f(d)notes the penalty function and has the following form:

f(d) = dF (d),

here the scalar function F (d) is the bulk part of the elastic energy. A typical example is to choose(d) as the GinzburgLandau penalization thus yielding the penalty function f(d) as:

F (d) = 14 20

(|d|2 1)2, f(d) = 1 20

(|d|2 1)d,here 0 > 0 is a constant.Let R3 be a bounded smooth domain. In this paper, we will consider the following initialundary conditions:

(,u,d)t=0 =

(0(x),m0(x),d0(x)

), x , (1.2)

d

u| = 0, d| = d0(x), x , (1.3)

here

0 L (), 0 0; d0 L() H1();

m0 L1(), m0 = 0 if 0 = 0; |m0|2

0 L1().

The global existence of weak solutions (,u,d) to the initialboundary problem (1.1)(1.3) in threemensions with > 3/2 was obtained by Wang and Yu in [29] and Liu and Qing in [26], indepen-

ntly. We should also remark that, when system (1.1) is incompressible, Jiang and Tan [13] and Liu

Y.-F. Yang et al. / J. Differential Equations 255 (2013) 12331253 1235

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w

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cationepooeqStsysufotoheWd Zhang in [25] proved the global weak existence of solutions to the ow of nematic liquid crys-ls for uids with non-constant density. Based on the existence result, Dai et al. in [1] extended thegularity and uniqueness results of Lin and Liu in [19] to the systems of nematic liquid crystals withn-constant uid density. Since the global existence of the weak solutions has been established, it istural to study their uniqueness property. To our best knowledge, so far there are very few resultsncerning uniqueness of weak solutions to the initialboundary value problem (1.1)(1.3).When the OssenFrank energy conguration functional reduces to the Dirichlet energy functional,

e hydrodynamic ow equation of nematic compressible liquid crystals can actually be written asllows:

t + div(u) = 0,(u)t + div(u u) + P = u div

(d d 1

2|d|2I3

),

dt + u d = (d+ |d|2d),

(1.4)

here d S2 and the other symbols have the same meanings with those in system (1.1). We refer toe readers to consult the recent papers [4] and [26] for the derivation of the system (1.4). For thestem (1.4), we are concerned with the same initial conditions (1.2) but with the following differentundary conditions:

u| = 0, d

= 0, x , (1.5)

here is the unit outer normal vector of .In contrast with system (1.1), from the mathematical point of view, it is much more dicult to deal

ith the nonlinear term |d|2d appearing in the third equation of (1.4). Even for the incompressiblew, there have been no satisfactory results concerning the global existence of weak solutions. Re-ntly, Lin et al. in [21] proved the existence of global-in-time weak solutions on a bounded smoothmain in R2. For three-dimensional case, the problem is still open. We should mention that Li andang in [16] have established its weakstrong uniqueness principle in three dimensions, providedat the existence of its weak solution is obtained.The compressible ow (1.4) of liquid crystals is much more complicated and hard to study math-atically due to the compressibility. For the one-dimensional case, the global existence of smoothd weak solutions to the compressible ow of liquid crystals was obtained by Ding et al. in [2,3].for three-dimensional case, Huang et al. in [12] and Liu and Zhang in [25] established the localistence of a unique strong solution provided that the initial data are suciently regular and satisfynatural compatibility condition. However, the global existence of weak solution to the compressiblew of liquid crystals in multi-dimensions is still open.We would like to point out that the system (1.4) includes several important equations as specialses: (i) When is constant, the rst equation in (1.4) reduces to the incompressibility condi-n of the uid (divu = 0), and the system (1.4) becomes the equation of incompressible ow ofmatic liquid crystals provided that P is an unknown pressure function. This was previously pro-sed by Lin [17] as a simplied EricksenLeslie equation modeling incompressible liquid crystalws. (ii) When d is a constant vector eld, the system (1.4) becomes a compressible NavierStokesuation. There have been a number of papers in the literature on the multi-dimensional Navierokes equations (see [6,24,27] and the references therein). (iii) When both and d are constants, thestem (1.4) becomes the incompressible NavierStokes equation provided that P is an unknown pres-re function, the fundamental equation to describe Newtonian uids (see Lions [23] and Temam [28]r survey of important developments). (iv) When is constant and u = 0, the system (1.4) reducesthe equation for the heat ow of harmonic maps into S2. There have been extensive studies on theat ow of harmonic maps in the past few decades (see, for example, the monograph by Lin and

ang [22] and the references therein).


toisfeintiesexticoatglrestdaentitoneNunThclcoww

LebaCow(fanmththeffrisde

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thIt is well known that the uniqueness of weak solutions is another fundamental and importantpic in mathematical theory of the compressible NavierStokes equations. In contrast with the ex-tence theory of weak solutions, it seems that uniqueness issue is more dicult and there are onlyw results dealing with uniqueness. With the aid of concept of the relative entropy, Germain [11]troduced a class of (weak) solutions to the compressible NavierStokes system satisfying a rela-ve entropy inequality with respect to a (hypothetical) strong solution of the same problem, andtablished the weakstrong uniqueness property within this class (see also [10]). Unfortunately, theistence of weak solutions belonging to this class, where, in particular, the density possesses a spa-al gradient in a suitable Lebesgue space, is not known. Recently, Feireisl et al. [7] introduced thencept of suitable weak solution for the compressible NavierStokes system, satisfying a general rel-ive entropy inequality with respect to any suciently regular pair of functions. They showed theobal-in-time existence of the suitable weak solutions for any nite energy initial data. Moreover, thelative entropy inequality can be used to show that suitable weak solutions comply with the weakrong uniqueness principle, meaning a weak and a strong solution emanating from the same initialta coincide as long as the latter exists. More recently, Feireisl et al. [8] further showed that any niteergy weak solution satises a relative entropy inequality with respect to any couple of smooth func-ons satisfying relevant boundary conditions. Based on the relative entropy inequality, they succeededprove the weakstrong uniqueness and to provide a satisfactory answer to the weakstrong unique-ss problem initially related to the fundamental questions of the well-posedness for the compressibleavierStokes system. In addition, Feireisl and Novotn [9] also established the similar weakstrongiqueness property of the variational solutions for the compressible NavierStokesFourier system.e authors of the present paper [30] have established the weakstrong uniqueness property in theass of nite energy weak solutions to the magnetohydrodynamic equations of three-dimensionalmpressible isentropic ows with the adiabatic exponent > 1 and constant viscosity coecients,hich improves the corresponding results for the compressible isentropic NavierStokes equationsith > 32 by Feireisl, Jin, and Novotn [8].This paper aims to establish the weakstrong uniqueness property for two simplied Ericksenslie systems (1.1) and (1.4), respectively, in the spirit of Feireisl et al. [8]. Our method is essentiallysed on the relative entropy, the modied relative entropy inequality, and a Gronwall type argument.mpared with the existence result of weak solution in [26,29], where both of them require > 3/2,e are going to make use of the techniques, established in [30], to estimate the remainder R (or R1)or the denition see (3.10) (or (4.10))), so as to establish the weakstrong uniqueness property forimproved lower bound for any adiabatic exponent > 1. When dealing with the compressible ne-

atic liquid crystal ow (1.1) or (1.4), the main diculty lies in the coupling and interaction betweene velocity eld u and the direction eld d. In particular, we should emphasize here that, as far ase weakstrong uniqueness property for the initialboundary problem (1.1)(1.3) is concerned, moreforts are required to build up the relative entropy inequality so as to overcome the effects arisingom inhomogeneous boundary condition d| = d0(x), which is quite different from the case of theentropic compressible NavierStokes equations investigated by Feireisl et al. in [8]. According to thenition of weak solutions for the compressible ow of liquid crystals, it is easy to see that

d L2(0, T ; H2()),the weak solution d actually solves the third equation of (1.1) (or (1.4)) in the strong sense. Conse-ently, we can combine the arguments of Feireisl et al. in [8] with the energy estimates for parabolicuation to obtain the desired relative entropy inequality.The sizes of the positive constants , , and do not play important roles in our proofs, we shall

erefore assume, for simplicity, that

= = = 1, (1.6)roughout this paper. In addition, to simplify the notations, we always set


in

anmTh

2.

leA (B)C = ((C )B) A,which A, B, and C are vectors in R3.The rest of the paper is organized as follows. In the next section, we recall the denitions of weakd strong solutions to the compressible ow of liquid crystals for two kinds of models and state theain results. Section 3 is devoted to the derivation of the relative entropy inequality and the proof ofeorem 2.1. Finally, we prove Theorem 2.2 in Section 4.

Main results

We say that a triplet {,u,d} is a nite energy weak solution to the initialboundary value prob-m (1.1)(1.3), if, for any T > 0,

0, L(0, T ; L ()), u L2(0, T ; H10()), u C0([0, T ], L2+1 () w),

d L((0, T ) ) L(0, T ; H1()) L2(0, T ; H2()),with (,u,d)(0, x) = (0(x),m0(x),d0(x)) for x .

The rst equation in (1.1) is replaced by a family of integral identities

( , )( , )dx

0(0, )dx =

0

(t + u )dxdt (2.1)

for any C1([0, T ] ), and any [0, T ]. Momentum equations (1.1)2 are satised in the sense of distributions, specically,

u( , )( , )dx

0u0 (0, )dx

=

0

(u t + u u : + P ()div u :

)dxdt

+

0

(d d

(1

2|d|2 + F (d)

)I3

): dxdt (2.2)

for any C1([0, T ] ), | = 0, and any [0, T ]. Eqs. (1.1)3 are replaced by a family of integral identities

d( , ) ( , )dx

d0 (0, )dx

=

0

(d t d : (d)u f(d)

)dxdt (2.3)

1for any C ([0, T ] ), | = 0, and any [0, T ].


lea

(0

anre

Thththda

pr The energy inequality

E(t) +

0

(|u|2 + d f(d)2)dxdt E(0) (2.4)holds for a.e. [0, T ], where

E(t) =

(1

2|u|2 + a

1 + 1

2|d|2 + F (d)

)dx,

and

E(0) =

(1

2

|m0|20

+ a 1

0 +

1

2|d0|2 + F (d0)

)dx.

The existence of global-in-time nite energy weak solutions to the initialboundary prob-m (1.1)(1.3) with the adiabatic exponent > 32 was established in [26,29], provided there existspositive constant C0 such that d f(d) 0 for all |d| C0 > 0.{, u, d} is called a classical (strong) solution to the initialboundary problem (1.1)(1.3) in

, T ) if{

C1([0, T ] ), (t, x) > 0 for all (t, x) (0, T ) ,u, t u,2u C

([0, T ] ), d, t d,2d C([0, T ] ) (2.5)d , u, d satisfy Eq. (1.1), together with the boundary conditions (1.3). Observe that hypothesis (2.5)quires the following regularity properties of the initial data:

{(0, ) = 0 C1(), 0 > 0,u(0, ) = u0 C2(), d(0, ) = d0 C2().

(2.6)

We are now ready to state the rst result of this paper.

eorem 2.1. Let R3 be a bounded domain with a boundary of class C2+ , > 0, and > 1. Supposeat {,u,d} is a nite energy weak solution to the initialboundary problem (1.1)(1.3) in (0, T ) ine sense specied above, and suppose that {, u, d} is a strong solution emanating from the same initialta (2.6).Then

, u u, d d.

In a similar way, we dene the nite energy weak solution {1,u1,d1} to the initialboundary valueoblem (1.4), (1.2), and (1.5) in the following sense: for any T > 0,

1 0, 1 L(0, T ; L ()), u1 L2(0, T ; H10()),

d1 L((0, T ) ) L(0, T ; H1()) L2(0, T ; H2()),with (1,1u1,d1)(0, x) = (0(x),m0(x),d0(x)) for x .


Re

Rethet

(1

ansi

Thth(0sa Similarly to (2.1)(2.3), Eqs. (1.4) hold in D((0, T ) ). The energy inequality

E1(t) +

0

(|u1|2 + d1 + |d1|2d12)dxdt E1(0) (2.7)

holds for a.e. [0, T ], where

E1(t) =

(1

21|u1|2 + a

11 +

1

2|d1|2

)dx,

and

E1(0) =

(1

2

|m0|20

+ a 1

0 +

1

2|d0|2

)dx.

mark 2.1. In the derivation of energy inequality (2.7), we have used the fact that |d1| = 1 to get

(td1 + u1 d1) |d1|2d1 = 12|d1|2

(t |d1|2 + u1 |d1|2

)= 0.mark 2.2. We should remark that the global-in-time renormalized nite energy weak solutions toe initialboundary value problem (1.4), (1.2) and (1.5) is still open. For one-dimensional case, Dingal. in [2,3] obtained the global-in-time existence of weak solutions.

Similarly, {1, u1, d1} is called a classical (strong) solution to the initialboundary value problem.4), (1.2) and (1.5) in (0, T ) if

{1 C1

([0, T ] ), 1(t, x) > 0 for all (t, x) (0, T ) ,u1, t u1,2u1 C

([0, T ] ), d1, t d1,2d1 C([0, T ] ) (2.8)d 1, u1, d1 satisfy Eqs. (1.4), together with the boundary conditions (1.5). Observe that hypothe-s (2.8) requires the following regularity properties of the initial data:

{1(0, ) = 0 C1(), 0 > 0,u1(0, ) = u0 C2(), d1(0, ) = d0 C2().

(2.9)

We now end up this section with another result of this paper.

eorem 2.2. Let R3 be a bounded domain with a boundary of class C2+ , > 0, and > 1. Supposeat {1,u1,d1} is a nite energy weak solution to the initialboundary value problem (1.4), (1.2) and (1.5) in, T ) in the sense specied above, and suppose that {1, u1, d1} is a strong solution emanating from theme initial data (2.9).Then

1 1, u1 u1, d1 d1.


3.

E

w

tofuso

wreovarin

th

an

SiProof of Theorem 2.1

Motivated by the concept of relative entropy in [8], we rst dene relative entropy E =([,u,d]|[, u, d]), with respect to {, u, d}, as

E =

(1

2|u u|2 + () ()( ) () + 1

2|d d|2

)dx, (3.1)

here

() = a 1

. (3.2)

In this section, we are going to deduce a relative entropy inequality satised by any weak solutionthe initialboundary value problem (1.1)(1.3). To this end, consider a triplet {, u, d} of smoothnctions, bounded away from zero in [0, T ] , u| = 0, and d| = d0(x). In addition, u and dlve the third equation of (1.1).Noticing that the boundary conditions for d and d are inhomogeneous, i.e. d| = d| = d0(x),

e should adapt and modify the arguments in [8] to build up the relative entropy inequality. Moregularity of d allows us to make use of energy estimates for parabolic equation, which help usercome the diculty due to the inhomogeneous boundary conditions. Consequently, combining theguments in [8] with the energy estimates for parabolic equations yields the desired relative entropyequality.To begin with, we take u as a test function in the momentum equation (2.2) to obtain

u u( , )dx

0u0 u(0, )dx

=

0

(u t u+ u u : u+ P ()div u u : u

)dxdt

0

((d f(d)) (d)u)dxdt. (3.3)

Second, we can use the scalar quantity = 12 |u|2 and = (), respectively, as test functions ine continuity equation (2.1) to get

1

2|u|2( , )dx =

1

20|u|2(0, )dx+

0

(u t u+ u (u)u

)dxdt (3.4)

d

()( , )dx =

0()(0, )dx+

0

(t

() + u ())dxdt. (3.5)nce (u,d) solves the third equation of (1.1) in the strong sense, it is easy to see that

( )

t(d d) + u d u d = (d d) f(d) f(d) , a.e.


M

Fiultiplying the above equation by (d d) and integrating over (0, ), we have

1

2|d d|2( , )dx+

0

|d d|2 dxdt

=

1

2

d0 d(0, )2 dx+

0

(d d) (d)udxdt

0

(d d) (d)udxdt +

0

(f(d) f(d)) (d d)dxdt. (3.6)

nally, multiplying the third equation of (1.1) by d f(d) gives

(1

2|d|2 + F (d)

)( , )dx+

0

d f(d)2 dxdt

=

(1

2|d0|2 + F (d0)

)dx+

0

(d f(d)) (d)udxdt. (3.7)

Summing up the relations (3.3)(3.6) with the energy inequality (2.4), we infer that

(1

2|u u|2 + () () + 1

2|d d|2

)( , )dx

+

0

|u u|2 dxdt +

0

|d d|2 dxdt

(1

20u0 u(0, )2 + (0) 0 ((0, ))+ 1

2

d0 d(0, )2)dx

+

0

(t u+ u u) (u u)dxdt +

0

u : (u u)dxdt

0

(t

() + u ())dxdt

0

P ()div udxdt

0

(d f(d)) (d)udxdt +

0

(d f(d)) (d)udxdt

+

0

(d d) ((d)u (d)u)dxdt +

0

(d d) (f(d) f(d))dxdt,

(3.8)


w

an

A

an

w

an

itprhere we have used (3.7). By virtue of the denition (3.2) of , it is easy to see that

() () = P ()d

(t

() + () u+ P ()div u)dx =

t P ()dx.

s a consequence, we deduce from the identities

P ()( , )dx

P ()(0, )dx =

0

t P ()dxdt

d (3.8) that the desired relative entropy inequality holds:

E( ) +

0

|u u|2 dxdt +

0

|d d|2 dxdt E(0) +

0

R(,u,d, , u, d)dt,

(3.9)

here

R = R(,u,d, , u, d) :=Rd +Rc, (3.10)Rd :=

(u u) (t u+ (u)u)dx+

u : (u u)dx

+

(( )t () + () (u u)

)dx

div u(P () P ())dx,

(3.11)

d

Rc : =

0

(d f(d)) (d)udxdt +

0

(d f(d)) (d)udxdt

+

0

(d d) ((d)u (d)u)dxdt +

0

(d d) (f(d) f(d))dxdt.(3.12)

In what follows, we shall nish the proof of Theorem 2.1 by applying the relative entropy inequal-y (3.9) to {, u, d}, where {, u, d} is a classical (smooth) solution of the initialboundary valueoblem (1.1), (2.6), and (1.3), such that(0, ) = 0, u(0, ) = u0, d(0, ) = d0.


w

Oby

tew

Af

CoAccordingly, the integrals depending on the initial data on the right-hand side of (3.9) vanish, ande apply a Gronwall type argument to deduce the desired result, namely,

, u u, d d.ur purpose is to examine all terms in the remainder (3.10) and to show that they can be absorbedthe left-hand side of (3.9).Compared with [8], we should remark that the main diculty comes from the coupling and in-

raction between the velocity eld u and the direction eld d. Moreover, in the context of theeakstrong uniqueness, we only assume the adiabatic exponent > 1, but not > 3/2 as in [8].Similarly to the proof of Theorem 2.1 in [30] (see also [8]), we use (3.11) to nd that

Rd =

(u u) ((u)(u u))dx+

( )

u (u u)dx

div u(P () P ()( ) P ())dx

div

(d d

(1

2|d|2 + F (d)

)I3

) (u u)dx

=

(u u) ((u)(u u))dx

div u(P () P ()( ) P ())dx

+

( )

[u div

(d d

(1

2|d|2 + F (d)

)I3

)] (u u)dx

(d f(d)) (d)(u u)dx

=:Rd

(d f(d)) (d)(u u)dx. (3.13)

ter a tedious but straightforward computation, it follows from (3.12) that

Rc :=Rc

(d f(d)) (d)(u u)dx

=

(d d) (f(d) f(d))dx+

(d d) (d d)udx

+

d (d d)(u u)dx+

f(d) (d d)(u u)dx

+

(f(d) f(d)) d(u u)dx. (3.14)

nsequently, it follows from the denitions of Rd and Rc thatR(,u,d, , u, d) =Rd +Rc . (3.15)


foth

H

O

Si

wth

foIn order to prove the weakstrong uniqueness property, we have to estimate the remainder R. Asr Rd , since the procedures are almost same as that in [30] (see also [8]), we just follow [30], liste outlines, and skip the details. In the sequel, we are going to focus on the estimation of Rc .From (2.6), it is clear to see that

(u u) ((u)(u u))dx

div u(P () P ()( ) P ())dx

CuL()E([,u,d][, u, d]). (3.16)

ere and hereafter C stands for a generic constant, which may change from line to line.Let

g= g(u, d,d,d) = u div(d d

(1

2|d|2 + F (d)

)I3

).

bviously, we have

1

( )g (u u)dx =

{ 2


fo

an

w

an

fo

for any > 0. On the other hand, noticing that

() ()( ) () C , as 2 2d

12 2 C, as 2 2 and > 1,e conclude from the denition of relative entropy E([,u,d]|[, u, d]) that

{2}

1

( )g (u u)dx

=

{2}

12 g 12 |u u|dx

=

{2}

( 12 2

)

2 g 12 |u u|dx

CgL()(

dx

) 12(

2|u u|2 dx

) 12

CgL()E([,u,d][, u, d]). (3.20)

Next, we continue to estimate Rc . Noticing that

dL((0,T )),dL((0,T )) C (3.21)d the fact that f is smooth, we have

(d d) (f(d) f(d))dx C(

|d d|2 dx) 1

2(

|d d|2 dx) 1

2

d d2L2() + C()d d2L2() d d2L2() + C()d d2L2() d d2L2() + C()E

([,u,d][, u, d]) (3.22)r any > 0, where we have used Sobolevs inequality. It follows from Hlders inequality that

(d d) (d d)udx uL()d dL2()d dL2() d d2L2() + C()u2L()d d2L2() d d2L2() + C()u2L()E

([,u,d][, u, d])(3.23)r any > 0. It is clear that


Fi

de

w

Iteq

4.

valiklintahaplthE1

d (d d)(u u)dx+

f(d) (d d)(u u)dx

(dL() + f(d)L())u uL2()d dL2()

u u2L2() + C()(dL() + f(d)L())2d d2L2()

u u2L2() + C()(dL() + f(d)L())2E([,u,d][, u, d]). (3.24)

nally, (3.21) and the fact that f is smooth imply that

(f(d) f(d)) (d)(u u)dx

CdL()u uL2()d dL2() u u2L2() + C()

(dL())2d d2L2() u u2L2() + C()

(dL())2E([,u,d][, u, d]). (3.25)Summing up relations (3.9)(3.25), we conclude that the relative entropy inequality yields thesired conclusion

E([,u,d][, u, d])( )

0

h(t)E([,u,d][, u, d])(t)dt,

here

h(t) = C{uL() +

g2

L3()+ gL() + u2L()

+ (dL() + f(d)L())2 + d2L() + 1}.

follows from (3.21) that h L1(0, T ). Thus, Theorem 2.1 immediately follows from Gronwalls in-uality.

Proof of Theorem 2.2

This section is devoted to proving the weakstrong uniqueness property for the initialboundarylue problem (1.4), (1.2), and (1.5). Compared with the problem discussed in Section 3, we woulde to point out two points: (i) The term |d1|2d1 in the third equation of (1.4) has higher non-earity than f(d) in the third equation of (1.1), so that more regularity results should be ob-ined to overcome the diculty caused by |d1|2d1. (ii) Different from d| = d0(x), we nowve homogeneous Neumann boundary condition for d1 at hand, namely,

d1 | = 0, which im-

ies that a modied relative entropy is required to prove the weakstrong uniqueness property fore initialboundary value problem (1.4), (1.2), and (1.5). To be precise, we dene relative entropy

= E1([1,u1,d1]|[1, u1, d1]), with respect to {1, u1, d1}, as


wva1th

m

Setin

an

Nw

MwE1 =

(1

21|u1 u1|2 +

((1) (1)(1 1) (1)

)

+ 12

(|d1 d1|2 + |d1 d1|2))dx, (4.1)

here is dened by (3.2), {1,u1,d1} is the nite energy weak solution to the initialboundarylue problem (1.4), (1.2), and (1.5) in the sense of Section 2, and {1, u1, d1} are smooth functions,bounded away from zero in [0, T ] , u1| = 0, and d1 | = 0. In addition, u1 and d1 solve

e third equation of (1.4).We rst establish the relative entropy inequality. For this purpose, take u1 as a test function in the

omentum equation to obtain

1u1 u1( , )dx

0u0 u1(0, )dx (4.2)

=

0

(1u1 t u1 + 1u1 u1 : u1 + P (1)div u1 u1 : u1

)dxdt

0

d1 (d1)u1 dxdt.

cond, we can use the scalar quantity 12 |u1|2 and (1), respectively, as test functions in the con-uity equation to get

1

21|u1|2( , )dx =

1

20|u1|2(0, )dx+

0

(1u1 t u1 + 1u1 (u1)u1

)dxdt

(4.3)

d

1(1)( , )dx =

0(1)(0, )dx+

0

(1t

(1) + 1u1 (1))dxdt.

(4.4)

otice that (u1,d1) solves the third equation of (1.4) in the strong sense, we have for almost every-here

t(d1 d1) + u1 d1 u1 d1 = (d1 d1) +(|d1|2d1 |d1|2d1).

ultiplying the above equation by (d1 d1) and (d1 d1), respectively, and integrating by parts,

e obtain


an

Si

1

2|d1 d1|2 dx+

0

|d1 d1|2 dxdt

=

1

2

d0 d1(0, )2 dx+

0

(|d1|2d1 |d1|2d1) (d1 d1)dxdt

0

((d1)u1 (d1)u1

) (d1 d1)dxdt (4.5)

d

1

2|d1 d1|2 dx+

0

|d1 d1|2 dxdt

=

1

2

d0 d1(0, )2 dx

0

(|d1|2d1 |d1|2d1) (d1 d1)dxdt

+

0

((d1)u1 (d1)u1

) (d1 d1)dxdt. (4.6)

milarly, we multiply the third equation of (1.4) by d1 + |d1|2d1 to obtain

1

2|d1|2( , )dx+

0

d1 + |d1|2d12 dxdt

=

1

2|d0|2 dx+

0

d1 (d1)u1 dxdt. (4.7)

Summing up the relations (4.2)(4.6) with the energy inequality (2.7), we infer that

(1

2|u1 u1|2 + (1) 1 (1) + 1

2

(|d1 d1|2 + |d1 d1|2))( , )dx

+

0

|u1 u1|2 dxdt +

0

|d1 d1|2 dxdt +

0

|d1 d1|2 dxdt

(1

20u0 u1(0, )2 + (0) 0 (1(0, ))

+ 1 (d d (0, )2 + d d (0, )2))dx

2

0 1 0 1


wob

w

an

R+

0

1(t u1 + u1 u1) (u1 u1)dxdt +

0

u1 : (u1 u1)dxdt

0

(1t

(1) + 1u1 (1))dxdt

0

P (1)div u1 dxdt

+

0

d1 (d1)(u1 u1)dxdt +

0

(d1 d1) ((d1)u1 (d1)u1

)dxdt

0

(d1 d1) (|d1|2d1 |d1|2d1)dxdt

+

0

(d1 d1) (|d1|2d1 |d1|2d1)dxdt

0

(d1 d1) ((d1)u1 (d1)u1

)dxdt, (4.8)

here we have used (4.7). Thus, we proceed the similar procedures as in the previous section totain the desired relative entropy inequality

E1( ) +

0

|u1 u1|2 dxdt +

0

(|d1 d1|2 + |d1 d1|2)dxdt

E1(0) +

0

R1(1,u1,d1, 1, u1, d1)dt, (4.9)

here

R1 = R1(1,u1,d1, 1, u1, d1) :=R1d +R1c, (4.10)R1d :=

1(u1 u1) ((u1)(u1 u1)

)dx

div u1(P (1) P (1)(1 1) P (1)

)dx

+

(1 1)1

[u1 div

(d1 d1 1

2|d1|2I3

)] (u1 u1)dx (4.11)

d

1c := (

d1 (d1) d1 (d1))(u1 u1)dx+

(d1 d1)

((d1)u1 (d1)u1

)dx


Sitiin

Aap

Toca

Le

O

Si

fo

(d1 d1) (|d1|2d1 |d1|2d1)dx+

(d1 d1) (|d1|2d1 |d1|2d1)dx

(d1 d1) ((d1)u1 (d1)u1

)dx. (4.12)

milar to the proof of Theorem 2.1, we shall nish the proof of Theorem 2.2 by applying the rela-ve entropy inequality (4.9) to {1, u1, d1}, where {1, u1, d1} is a classical (smooth) solution of theitialboundary value problem (1.4), (2.6), and (1.5), such that

1(0, ) = 0, u1(0, ) = u0, d1(0, ) = d0.s a result, the integrals depending on the initial data on the right-hand side of (4.9) vanish, and weply a Gronwall type argument to deduce the desired result, namely,

1 1, u1 u1, d1 d1.complete the proof, we have to examine all terms in the remainder (4.10) and to show that theyn be absorbed by the left-hand side of (4.9).From (2.6), it is clear to see that

1(u1 u1) ((u1)(u1 u1)

)dx

div u1(P (1) P (1)(1 1) P (1)

)dx

Cu1L()E1. (4.13)

t

g1 = g1(u1,d1,d1) = u1 div(d1 d1 1

2|d1|2I3

).

bviously, we have

1

1(1 1)g1 (u1 u1)dx

=

{01


w

an

Fo

fo

an

foNow we are going to estimate R1c . From (4.12), a direct calculation gives

R1c =Ra1c +Rb1c (4.17)

here

Ra1c =

(d1 d1) (d1 d1)u1 dx+

d1 (d1 d1)(u1 u1)dx

(|d1| + |d1|)d1 (d1 d1)(|d1| |d1|)dx+

(d1 d1) (d1)(u1 u1)dx (4.18)

d

Rb1c =

(d1 d1) (d1 d1)u1 dx+

(|d1| + |d1|)d1 (d1 d1)(|d1| |d1|)dx+

|d1 d1|2|d1|2 dx. (4.19)

r the rst term on the right-hand side of (4.18), we have

(d1 d1) (d1 d1)u1 dx

u1L()d1 d1L2()d1 d1L2() d1 d12L2() + C()u12L()d1 d12L2() d1 d12L2() + C()u12L()E1, (4.20)

r any > 0. In a similar way, we also obtain

d1 (d1 d1)(u1 u1)dx u1 u12L2() + C()d12L()E1, (4.21)

(|d1| + |d1|)d1 (d1 d1)(|d1| |d1|)dx

d1 d12L2() + C()(d12L() + d12L())d12L()E1, (4.22)

d

(d1 d1) (d1)(u1 u1)dx u1 u12L2() + C()d12L()E1, (4.23)r any > 0.


Li

an

Se

w

w

Resoapliq

A

thw(GtoNFinally, the Cauchy inequality and the denition of the relative entropy E1 give

(d1 d1) (d1 d1)u1 dx u1L()d1 d1L2()d1 d1L2() u1L()E

121 E

121

u1L()E1. (4.24)

kewise, we get

(|d1| + |d1|)d1 (d1 d1)(|d1| |d1|)dx

d1L()(d1L() + d1L())d1 d1L2()d1 d1L2()

d1L()(d1L() + d1L())E1 (4.25)

d

|d1 d1|2|d1|2 dx d12L()d1 d12L2() d12L()E1. (4.26)

Moreover, by applying the quasilinear equations of parabolic type estimates (see [14, Chapter VI,ction 3]) to the third equation of (1.4) and a density argument, we have

d1L() C . (4.27)

Summing up relations (4.10)(4.27) with the denition of nite energy weak solution {1,u1,d1},e conclude from the relative entropy inequality (4.9) that

E1([1,u1,d1][1, u1, d1])( )

0

h1(t)E1([1,u1,d1][1, u1, d1])(t)dt,

here h1 L1(0, T ). Thus, Theorem 2.1 immediately follows from Gronwalls inequality.

mark 4.1. For one-dimensional case, Ding et al. in [2,3] established the existence of global classicallutions and the existence of global weak solutions. Based on the existence results, one can directlyply Theorem 2.2 to obtain the weakstrong uniqueness principle for one-dimensional compressibleuid crystal system.

cknowledgments

The authors are grateful to the referee for valuable suggestions. The authors also would like toank Prof. Song Jiang for helpful discussions and sustained encouragement. Yong-Fu Yangs researchas in part supported by NSFC (Grant No. 11201115) and China Postdoctoral Science Foundationrant No. 2012M510365). Changsheng Dous research was partially supported by China Postdoc-ral Science Foundation (Grant No. 2012M520205). Ju Qiangchang was supported by NSFC (Grant

o. 11171035).


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[2

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Weak-strong uniqueness property for the compressible ow of liquid crystals1 Introduction2 Main results3 Proof of Theorem 2.14 Proof of Theorem 2.2AcknowledgmentsReferences

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Weakstrong Uniqueness Property for the Compressible Flow

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